Local Expansion of Symmetrical Graphs

Abstract

A graph is vertex-transitive (edge-transitive) if its automorphism group acts transitively on the vertices (edges, resp.). The expansion rate of a subset S of the vertex set is the quotient e(S):= |∂(S)|/|S|, where ∂(S) denotes the set of vertices not in S but adjacent to some vertex in S. Improving and extending previous results of Aldous and Babai, we give very simple proofs of the following results. Let X be a (finite or infinite) vertex-transitive graph and let S be a finite subset of the vertices. If X is finite, we also assume |S| ≤|V(X)/2. Let d be the diameter of S in the metric induced by X. Then e(S) ≥1/(d + 1); and e(S) ≥ 2/(d +2) if X is finite and d is less than the diameter of X. If X is edge-transitive then |δ(S)|/|S| ≥ r/(2d), where ∂(S) denotes the set of edges joining S to its complement and r is the harmonic mean of the minimum and maximum degrees of X. – Diverse applications of the results are mentioned.